Journal of Inequalities in Pure and
Applied Mathematics
NEWER APPLICATIONS OF GENERALIZED MONOTONE
SEQUENCES
volume 7, issue 5, article 167,
2006.
L. LEINDLER
Bolyai Institute, University of Szeged,
Aradi vértanúk tere 1
H-6720 Szeged, Hungary
Received 03 July, 2006;
accepted 06 December, 2006.
Communicated by: H. Bor
EMail: leindler@math.u-szeged.hu
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
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Abstract
A particular result of Telyakovskiı̌ is extended to the newly defined class of
numerical sequences and a specific problem is also highlighted. A further analogous result is also proved.
2000 Mathematics Subject Classification: 42A20, 40A05, 26D15.
Key words: Sine coefficients, Special sequences, Integrability.
The author was partially supported by the Hungarian National Foundation for Scientific Research under Grant # T042462 and TS44782.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Notions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
4
Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
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L. Leindler
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1.
Introduction
Recently several papers, see [4], [5] and [6], have dealt with the issue of uniform convergence and boundedness of monotone decreasing sequences. Further
results and extensions have also been reported by the author in [6].
In this paper we shall give two further results on boundedness for wider
classes of monotone sequences. First we present some theorems which will be
useful in the following sections of this paper. In Section 2 we state the main
results, in Section 3, we provide definitions and notations and in Section 4 we
give detailed proofs of the main theorem and corollary.
In [7] S.A. Telyakovskiı̌ proved the following useful theorem.
Theorem 1.1. If a sequence {nm } of natural numbers (n1 = 1 < n2 < n3 <
· · · ) is such that
Newer Applications of
Generalized Monotone
Sequences
L. Leindler
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(1.1)
∞
X
1
A
≤
n
nm
j=m j
Contents
for all m = 1, 2, . . . , where A > 1, then the estimate
−1
∞ nj+1
X
X
sin kx (1.2)
≤ KA
k
j=1 k=nj
holds for all x, where K is an absolute positive constant.
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−1
In [4], the author showed that the sequence {k } in (1.2) can be replaced
by any sequence c := {ck } which belongs to the class R0+ BV S.
J. Ineq. Pure and Appl. Math. 7(5) Art. 167, 2006
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Recently in [5] and [6] we verified as well that the sequence {k −1 } can
be replaced by sequences which belong to either of the classes γRBV S and
γGBV S.
More precisely we proved:
Theorem 1.2 (see [6]). Let γ := {γn } be a sequence of nonnegative numbers
satisfying the condition γn = O(n−1 ); furthermore let α := {αn } be a similar
sequence with the condition αn = o(n−1 ). If c := {cn } ∈ γGBV S, or belongs
to αGBV S, furthermore, if the sequence {nm } satisfies (1.1), then the estimates
(1.3)
−1
∞ nj+1
X
X
≤ K(c, {nm }),
c
sin
kx
k
j=1 k=nj
L. Leindler
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or
(1.4)
Newer Applications of
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Sequences
−1
∞ nj+1
X
X
= o(1),
c
sin
kx
k
j=m k=nj
Contents
m → ∞,
hold uniformly in x, respectively.
We note that, in general, (1.3) does not imply (1.4) see the Remark in [5].
We also note that, every quasi geometrically increasing sequence {nm } satisfies
the inequality (1.1) (see [3, Lemma 1]).
A consequence of Theorem 1.1 shows that not only series (1.2) but also
the Fourier series of any function of bounded variation possesses the property
analogous to (1.2) (see [7, Theorem 2]).
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Utilizing these results Telyakovskiı̌ [7] proved another theorem, which is an
interesting variation of a theorem by W.H. Young [8].
This theorem reads as follows.
Theorem 1.3. If the function f ∈ L(0, 2π) and the function g is of bounded
variation on [0, 2π], then the estimate
−1
∞ nj+1
X
X
≤ KAkf kL V (g)
(1.5)
(a
α
+
b
β
)
k
k
k
k
j=1 k=nj
is valid for any sequence {nm } with (1.1), where ak , bk and αk , βk are the
Fourier coefficients of f and g, respectively.
Newer Applications of
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L. Leindler
One can see that if we consider the function of bounded variation
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∞
π − x X sin kx
=
,
g(x) :=
2
k
k=1
0 < x < 2π,
then (1.5) reduces to
(1.6)
j=1 ∞ nj+1
X
X−1
k=nj
bk ≤ KAkf kL ,
k which strengthens the well-known result by H. Lebesgue [2, p. 102] that the
series
∞
X
bk
k
k=1
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converges for the functions f ∈ L(0, 2π).
These observations are made in [7] as well.
We have recalled (1.6) because one of our aims is to show that the sequence
{k −1 } appearing in (1.6) can be replaced, as was the case in (1.2), by any sequence {βk } ∈ γGBV S, if γn = O(n−1 ).
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2.
Results
We prove the following assertions.
Theorem 2.1. If the function f ∈ L(0, 2π) with {bk } Fourier sine coefficients,
the sequence {nm } is quasi geometrically increasing, and the sequence {βk }
belongs to γGBV S or αGBV S, where α and γ have the same definition as in
Theorem 1.2, then
−1
∞ nj+1
X
X
(2.1)
bk βk ≤ K({nm }, {βk })kf kL ,
j=1 k=nj
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L. Leindler
or
(2.2)
−1
∞ nj+1
X
X
= o(1),
b
β
k
k
j=m k=nj
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m → ∞,
hold, respectively.
Remark 1. It is clear that if a sine series with coefficients {βn } ∈ γGBV S
and γn = O(n−1 ), that is, if the function
g(x) :=
∞
X
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βk sin kx
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had a bounded variation, then (2.1) would be a special case of (1.5).
J. Ineq. Pure and Appl. Math. 7(5) Art. 167, 2006
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The author is unaware of such a result, or its converse. It is an interesting
open question.
Utilizing our result (2.1) and the method of Telyakovskiı̌ used in [7] we can
also obtain estimates for En (f )L and ων (f, δ)L .
Corollary 2.2. If f (x), γ, {bk }, {βk } and {nm } are as in Theorem 2.1, then
for any n with ni ≤ n < ni+1 the following estimates
1
(2.3)
ων f,
n L
≥ K(ν)En (f )L


−1
−1
∞ nj+1
ni+1
X
X
X
≥ K(ν, {nm }, {βk }) 
bk βk +
bk βk 
j=i+1 k=nj
k=n+1
hold.
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3.
Notions and Notations
A positive null-sequence c := {cn } (cn → 0) belongs to the family of sequences
of rest bounded variation, and briefly we write c ∈ R0+ BV S, if
∞
X
|∆ cn | ≤ Kcm ,
(∆ cn = cn − cn+1 ),
n=m
holds for all m ∈ N, where K = K(c) is a constant depending only on c.
In this paper we shall use K to designate either an absolute constant or a
constant depending on the indicated parameters, not necessarily the same at
each occurrence.
Let γ := {γn } be a given positive sequence. A null-sequence c of real
numbers satisfying the inequality
∞
X
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|∆ cn | ≤ K γm
n=m
is said to be a sequence of γ rest bounded variation, represented by c ∈ γRBV S.
If γ is a given sequence of nonnegative numbers, the terms cn are real and
the inequality
2m
X
|∆ cn | ≤ K γm ,
m = 1, 2, . . .
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n=m
holds, then we write c ∈ γGBV S.
The class γGBV S of sequences is wider than any one of the classes γRBV S
and GBV S. The class GBV S was defined in [1] by Le and Zhou with γm :=
max |cn |, where N is a natural number.
m≤n<m+N
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A sequence β := {βn } of positive numbers is called quasi geometrically
increasing (decreasing) if there exist natural numbers µ and K = K(β) ≥ 1
such that for all natural numbers n,
1
βn+µ ≥ 2βn and βn ≤ K βn+1
βn+µ ≤ βn and βn+1 ≤ K βn .
2
Let En (f )L denote the best approximation of the function f in the metric L
by trigonometric polynomials of order n; and tn (f, x) be a polynomial of best
approximation of f (x) in the metric L by trigonometric polynomials of order
n.
Finally denote by ων (f, δ)L the integral modulus of continuity of order ν of
f ∈ L.
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4.
Proofs
In this section we detail proofs of Theorem 2.1 and Corollary 2.2.
Proof of Theorem 2.1. It is clear that
Z nj+1 −1
nj+1 −1
X
1 2π X
bk βk =
βk sin kx dx.
π 0 k=n
k=n
j
j
Thus
nj+1
nj+1
Z
−1
−1
X
1 2π
X
dx.
b
β
≤
|f
(x)|
β
sin
kx
k
k
π
0
k=nj
k=nj
Let us sum up these inequalities and apply the estimate (1.3) with βk in place of
ck , we get that
nj+1 −1
nj+1 −1
Z
∞
∞
2π
X X
X X
1
dx
≤
|f
(x)|
β
sin
kx
b
β
k
k
k
π
0
j=1 k=nj
j=1 k=nj
≤ K({βk }, {nm })kf kL ,
which proves (2.1).
If we sum only from m to infinity and use the assertion (1.4) instead of (1.3),
we clearly obtain (2.2).
Herewith Theorem 2.1 is proved.
Proof of Corollary 2.2. It is easy to see that Jackson’s theorem and the estimate
(2.1) with f (x) − tn (f, x) in place of f (x) yield (2.3) immediately.
An itemized reasoning can be found in [7].
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References
[1] R.J. LE AND S.P. ZHOU, A new condition for uniform convergence of certain trigonometric series, Acta Math. Hungar., 10(1-2) (2005), 161–169.
[2] H. LEBESGUE, Leçons sur les Séries Trigonometriques, Paris: GauthierVillars, 1906.
[3] L. LEINDLER, On the utility of power-monotone sequences, Publ. Math.
Debrecen, 55(1-2) (1999), 169–176.
[4] L. LEINDLER, On the uniform convergence and boundedness of a certain
class of sine series, Analysis Math., 27 (2001), 279–285.
[5] L. LEINDLER, A note on the uniform convergence and boundedness of a
new class of sine series, Analysis Math., 31 (2005), 269–275.
[6] L. LEINDLER, A new extension of monotone sequences and its applications, J. Inequal. Pure and Appl. Math., 7(1) (2006), Art. 39.
[7] S.A. TELYAKOVSKIǏ, On partial sums of Fourier series of functions of
bounded variation, Proc. Steklov Inst. Math., 219 (1997), 372–381.
[8] W.H. YOUNG, On the integration of Fourier series, Proc. London Math.
Soc., 9 (1911), 449–462.
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